January 29, 2020

# Exam 4

Physics 207 – Introductory Physics I
Nazareth College
Department of Chemistry & Biochemistry
Robert F Szalapski, PhD – Adjunct Lecturer
www.CallMeDrRob.com
Fall 2011

Exam 4
Circular Motion, Gravitation, Work & Energy

## § Problem 1

Two stars of equal mass are gravitationally bound in a binary star system. When this happens they orbit the point which is midway between them. As shown in the picture, the two stars, due to their mutual gravitational attraction, will continue to move in the circular orbit shown. Let

The stars are the same size as our Sun, and the orbital radius is the same size as the mean Earth–Sun distance, but that’s just a fun coincidence.

### § (a)

What is the strength of the gravitational force felt by one star due to the other. Clearly indicate the equation used, then substitute the numbers, and finally present the numerical answer. (10 points)

Note that the distance between the two stars is twice the radius of the circle.

### § (b)

What will be the speed of either star. (It will be the same for both stars, so you only need to solve once.) (10 points)

We need to equate the force from part (a) to supply the centripetal force to keep the star moving on a circle of radius . I choose to go back to the expression for the force rather than just plugging in the number, but either approach would be okay.

### § (c)

What is the angular velocity of either star? (5 points)

### § (d)

What is the orbital period of either star? (The period is the time required to orbit once.) (5 points)

One orbit is and takes place in time .

## § Problem 2

On a theme-park thrill ride a roller coaster travels over the outside of a circular loop of radius with a speed of . A safety harness is used to hold each rider safely in the car. What force must the safety harness provide when the rider, mass , is exactly at the top of the loop? (20 points)

The normal force, the weight and the centripetal acceleration all point from the top of the circle towards the center of the circle. This is virtually identical to problems in the homework.

Incidentally

The numbers were chosen so that the force of the safety harness is equal to the person’s weight.

## § Problem 3

A regulation ping-pong ball with a mass of is shown in the figure. It is at rest on a massless compressed coil spring with a spring constant . The spring as it is shown is compressed by a distance of .
In the original configuration shown:

### § (a)

What is the initial elastic potential energy stored in the spring? (4 points)

### § (b)

What is the initial gravitational potential energy of the ping-pong ball? (4 points)

### § (c)

What is the initial kinetic energy of the ping-pong ball? (4 points)

The spring is allowed to expand pushing the ball upwards. When the spring has expanded to its equilibrium position the ball will separate from the spring.

### § (d)

What is the elastic potential energy stored in the spring at the instant the ball separates? (4 points)

### § (e)

What is the gravitational potential energy of the ping-pong ball at the instant it separates from the spring? (4 points)

### § (f)

What is the kinetic energy of the ping-pong ball at the instant it separates from the spring? (4 points)

First we should take note of the total energy in the system which we may compute from the original configuration.

At the instant of separation,

### § (g)

What is the maximum height reached by the ping-pong ball? (4 points)

Using conservation of energy

In this last step one could also use kinematic equations since there was not a requirement to use conservation of energy.

## § Problem 4

A skier approaches a ski jump which makes an angle of with the horizontal. The jump will lift the skier a height . The skier has a speed of when she reaches the start of the jump.

### § (a)

Use conservation of energy to find the speed as the skier leaves the jump assuming that friction is negligible. (9 points)

Using conservation of energy

### § (b)

Use conservation of energy to find the speed as the skier leaves the jump if the coefficient of kinetic friction is . (9 points)

This problem differs from part (a) in that some energy will be dissipated through friction. If is the magnitude of the work done by friction, that is, taken as a positive number, then we may modify the energy-conservation equation from part (a) as

This means that should come out smaller. To compute the work,

The distance is the hypotenuse of the triangle shown, so

Where I have used .

Ignoring the sign from the since we want the magnitude of the work,

We now substitute that into the energy-conservation equation.

Then

which is a bit less than the result found in part (a), as expected.